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22
votes
3answers
4k views

What is the difference between implicit and explicit time dependence e.g. $\frac{\partial \rho}{\partial t}$ and $\frac{d \rho} {dt}$?

What is the difference between implicit and explicit time dependence e.g. $\frac{\partial \rho}{\partial t}$ and $\frac{d \rho} {dt}$? I know one is a partial derivative and the other is a total ...
1
vote
4answers
182 views

Rotation systems. Problem interpreting an equation

In this equation: $$ \mathbf a_i\overset{\rm def}{=}\left(\frac{d^2\mathbf r}{dt^2}\right)_i=\left(\frac{d\mathbf ...
19
votes
7answers
9k views

Laplace operator's interpretation

What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in ...
10
votes
2answers
402 views

Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $

I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are ...
6
votes
5answers
847 views

What is the meaning of following expression $C=\frac{\delta Q}{dT}$ mathematically?

Our professor raised the following question during our lecture in Statistical Physics (even so it's related to Thermodynamics): Many text books (even Wikipedia) writes wrong expressions (from ...
4
votes
2answers
1k views

Derivative of the product of operators and Derivative of exponential

I'm asked to show that $$\frac{d(\hat{A}\hat{B})}{d\lambda} ~=~ \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$$ With $\lambda$ a continuous parameter. Should I use the ...
5
votes
7answers
764 views

Physical intuition for higher order derivatives

Could somebody give me an intuitive physical interpretation of higher order derivatives (from 2 and so on), that is not related to position - velocity - acceleration - jerk - etc?
1
vote
1answer
166 views

Total and partial derivatives in thermodynamics and Maxwell relations

Consider the expression $$dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV$$ I'm trying to understand how to derive an expression for $\left( ...
1
vote
1answer
1k views

What is the common difference between partial time derivative and ordinary time derivative? [duplicate]

What is difference between partial and ordinary time derivative? for example: what is difference between $\frac {\partial v}{\partial t}$ and $\frac {dv}{dt}$? where the $v$ is velocity.
7
votes
4answers
477 views

Conserved quantities and total derivatives?

I am having a bit of a crisis in understanding of the physical meanings of total derivatives. When a quantity $\rho$ (be it a vector or a scalar) is said to be conserved, then (mathematically) ...
-2
votes
1answer
75 views

Finding solution to this differential equation

In this paper http://arxiv.org/abs/hep-th/9506035 equation (3.11) was written as: $$\frac{\partial L}{\partial u}\frac{\partial L}{\partial v} = -1$$ The author then said p.9 that "approximate ...
3
votes
1answer
642 views

Time evolution in quantum mechanics

We know that an operator A in quantum mechanics has time evolution given by Heisenberg equation: $$ \frac{i}{\hbar}[H,A]+\frac{\partial A}{\partial t}=\frac{d A}{d t} $$ Can we derive from this ...
2
votes
4answers
989 views

Which Schrödinger equation is correct?

In the coordinate representation, in 1D, the wave function depends on space and time, $\Psi(x,t)$, accordingly the time dependent Schrödinger equation is $$H\Psi(x,t) = ...
1
vote
5answers
292 views

What is divergence?

What is divergence? I was learning about Maxwells equations and don't understand the divergence part of it. Can someone give an intuition of what divergence is in relation to maxwells equation. To ...
4
votes
2answers
2k views

Derivatives of operators

How do derivatives of operators work? Do they act on the terms in the derivative or do they just get "added to the tail"? Is there a conceptual way to understand this? For example: say you had the ...
14
votes
2answers
4k views

Difference between $\Delta$, $d$ and $\delta$

I have read the thread regarding 'the difference between the operators between $\delta$ and $d$', but it does not answer my question. I am confused about the notation for change in Physics. In ...
39
votes
3answers
3k views
4
votes
2answers
141 views

Physical meaning of harmonic function?

In complex numbers, we define a harmonic function as a twice continuously differentiable function such that the Laplace operator acting on it gives zero. Can anybody explain me the physical ...
5
votes
2answers
226 views

A confusion about notation in Goldstein

On treating systems of particles, Goldstein starts with the consideration that whenever there are $k$ particles on a system, the $i$-th one obeys the relation $$\dfrac{d}{dt}{\bf p}_i = {\bf ...
3
votes
1answer
258 views

What is a covariant derivative in gauge theory?

I've been studying electroweak theory and you need to keep the Lagrangian covariant by introducing covariant derivatives. What is a covariant derivative? And what does it mean to keep the Lagrangian ...
2
votes
4answers
258 views

Poincare invariant Lagrangians

The Lagrangian density of a Poincare invariant theory should not depend explicitly on the space-time coordinates. Does this mean $$ \partial_\mu \mathcal{L}=0~? $$ If this is the case doesn't the ...
0
votes
1answer
56 views

Geodesic equation proof confusing me

I was looking through this proof and have no idea where the $u$ comes from. Any help is appreciated. This is from here; I want to know how they got from eqn 5 to eqn 6.
0
votes
2answers
305 views

Feynman's subscript notation

Consider this vector calculus identity: $$ \mathbf{A} \times \left( \nabla \times \mathbf{B} \right) = \nabla_\mathbf{B} \left( \mathbf{A \cdot B} \right) - \left( \mathbf{A} \cdot \nabla \right) ...
1
vote
0answers
40 views

Partial derivatives in Lagrangian formalism [duplicate]

Suppose I have a function $f = xy$. A partial derivative of $f$ with respect to $x$ implies holding $y$ constant: $$ \frac{\partial f}{\partial x} = y $$ Does this mean that in order to evaluate ...
0
votes
1answer
22 views

Is difference in wave number always small?

Over the last few days I have been looking at a derivation of group velocity. The derivation is the one shown in this question Deriving group velocity. I have seen this derivation in many places, and ...
0
votes
1answer
109 views

In Newtonian pressure, what type of function is force?

This is pressure in Newtonian mechanics: $$P=\frac {dF}{dA}.$$ What does this mean? (Doesn't it mean that force is a function of area?) What type of function is force?